Random Walk in Python
Learn how to use Python to make a Random Walk
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A random walk can be thought of as a random process in which a token or a marker is randomly moved around some space, that is, a space with a metric used to compute distance. It is more commonly conceptualized in one dimension ($\mathbb{Z}$), two dimensions ($\mathbb{Z}^2$) or three dimensions ($\mathbb{Z}^3$) in Cartesian space, where $\mathbb{Z}$ represents the set of integers. In the visualizations below, we will be using scatter plots as well as a colorscale to denote the time sequence of the walk.
Random Walk in 1D¶
The jitter in the data points along the x and y axes are meant to illuminate where the points are being drawn and what the tendency of the random walk is.
import plotly.graph_objects as go
import numpy as np
np.random.seed(1)
l = 100
steps = np.random.choice([-1, 1], size=l) + 0.05 * np.random.randn(l) # l steps
position = np.cumsum(steps) # integrate the position by summing steps values
y = 0.05 * np.random.randn(l)
fig = go.Figure(data=go.Scatter(
x=position,
y=y,
mode='markers',
name='Random Walk in 1D',
marker=dict(
color=np.arange(l),
size=7,
colorscale='Reds',
showscale=True,
)
))
fig.update_layout(yaxis_range=[-1, 1])
fig.show()
Random Walk in 2D¶
import plotly.graph_objects as go
import numpy as np
l = 1000
x_steps = np.random.choice([-1, 1], size=l) + 0.2 * np.random.randn(l) # l steps
y_steps = np.random.choice([-1, 1], size=l) + 0.2 * np.random.randn(l) # l steps
x_position = np.cumsum(x_steps) # integrate the position by summing steps values
y_position = np.cumsum(y_steps) # integrate the position by summing steps values
fig = go.Figure(data=go.Scatter(
x=x_position,
y=y_position,
mode='markers',
name='Random Walk',
marker=dict(
color=np.arange(l),
size=8,
colorscale='Greens',
showscale=True
)
))
fig.show()
Random walk and diffusion¶
In the two following charts we show the link between random walks and diffusion. We compute a large number N
of random walks representing for examples molecules in a small drop of chemical. While all trajectories start at 0, after some time the spatial distribution of points is a Gaussian distribution. Also, the average distance to the origin grows as $\sqrt(t)$.
import plotly.graph_objects as go
import numpy as np
l = 1000
N = 10000
steps = np.random.choice([-1, 1], size=(N, l)) + 0.05 * np.random.standard_normal((N, l)) # l steps
position = np.cumsum(steps, axis=1) # integrate all positions by summing steps values along time axis
fig = go.Figure(data=go.Histogram(x=position[:, -1])) # positions at final time step
fig.show()
import plotly.graph_objects as go
from plotly.subplots import make_subplots
import numpy as np
l = 1000
N = 10000
t = np.arange(l)
steps = np.random.choice([-1, 1], size=(N, l)) + 0.05 * np.random.standard_normal((N, l)) # l steps
position = np.cumsum(steps, axis=1) # integrate the position by summing steps values
average_distance = np.std(position, axis=0) # average distance
fig = make_subplots(1, 2)
fig.add_trace(go.Scatter(x=t, y=average_distance, name='mean distance'), 1, 1)
fig.add_trace(go.Scatter(x=t, y=average_distance**2, name='mean squared distance'), 1, 2)
fig.update_xaxes(title_text='$t$')
fig.update_yaxes(title_text='$l$', col=1)
fig.update_yaxes(title_text='$l^2$', col=2)
fig.update_layout(showlegend=False)
fig.show()
Advanced Tip¶
We can formally think of a 1D random walk as a point jumping along the integer number line. Let $Z_i$ be a random variable that takes on the values +1 and -1. Let this random variable represent the steps we take in the random walk in 1D (where +1 means right and -1 means left). Also, as with the above visualizations, let us assume that the probability of moving left and right is just $\frac{1}{2}$. Then, consider the sum
Sn=n∑i=0Ziwhere S_n represents the point that the random walk ends up on after n steps have been taken.
To find the expected value
of $S_n$, we can compute it directly. Since each $Z_i$ is independent, we have
but since $Z_i$ takes on the values +1 and -1 then
E(Zi)=1⋅P(Zi=1)+−1⋅P(Zi=−1)=12−12=0Therefore, we expect our random walk to hover around $0$ regardless of how many steps we take in our walk.
What About Dash?¶
Dash is an open-source framework for building analytical applications, with no Javascript required, and it is tightly integrated with the Plotly graphing library.
Learn about how to install Dash at https://dash.plot.ly/installation.
Everywhere in this page that you see fig.show()
, you can display the same figure in a Dash application by passing it to the figure
argument of the Graph
component from the built-in dash_core_components
package like this:
import plotly.graph_objects as go # or plotly.express as px
fig = go.Figure() # or any Plotly Express function e.g. px.bar(...)
# fig.add_trace( ... )
# fig.update_layout( ... )
from dash import Dash, dcc, html
app = Dash()
app.layout = html.Div([
dcc.Graph(figure=fig)
])
app.run(debug=True, use_reloader=False) # Turn off reloader if inside Jupyter
